Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 3 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

November 25, 2023 at 10:04am

#1060085

I Might Have Overused the Word 'Doom'

Destiny, fate, and doom are roughly synonymous with each other. Only doom, however, conveys the appropriate sense of impending ruination.

‘It’s absolutely guaranteed’: the best and worst case scenarios for sea level rise

I used to think that "doom" was etymologically related to "damn," but this is why we look things up instead of going with our initial guesses.

Not only is dangerous sea level rise “absolutely guaranteed”, but it will keep rising for centuries or millennia even if the world stopped emitting greenhouse gases tomorrow, experts say.

Oi! You're not supposed to tell them that. It makes people wonder why they're bothering to do anything.

Rising seas are one of the most severe consequences of a heating climate that are already being felt.

Okay, but at least we know where that impact will be felt: near sea level. Other consequences can and will (and do) hit anywhere.

To stop the acceleration of sea level rise over the past century, Bamber says, we would have to go back to pre-industrial temperatures.

And the only way that's going to happen is if humanity gets (at least mostly) wiped out. Which, despite the confident claims of people even more pessimistic than I am, climate change probably won't do that by itself.

“The thing about sea level rise is that it is absolutely guaranteed,” Bamber says. “If you warm the planet, sea level is going to go up, period, no caveats. The oceans warm up and the ice melts. It’s an absolute given of global heating.”

Simple solution: build a space elevator. Better yet, build two, one in each of the major oceans. Use them to suck up the excess water and release it into space. As a bonus, Earth will finally get that set of rings it's jealous of Saturn for having.

The impact is hard to gauge because the ocean does not rise at the same speed uniformly, it’s not like a bath. For one thing, Earth is not a perfect sphere; temperatures are also different across the planet, and are affected by ocean currents.

This sort of thing, which I found out about long before reading this article, took me a while to wrap my head around. I mean, sea level is sea level, right? Sure, tides, yeah, but you can average those out and the result should be the same datum for every coast, right?

Wrong.

I think, but I'm not completely sure, that I came to this realization while learning about the Suez Canal. What? Don't look at me like that; I'm a civil engineer and, for once, I was learning about something actually related to my career. Apparently, originally, the Suez Canal flowed one way in summer and the other in winter, due to seasonal differences in sea level between the Mediterranean and Red Seas.

Point is, as the article notes, many different factors contribute to sea level at any given location.

So how bad could things get? Again, it’s hard to predict exactly, but the IPCC has tried its hand at modelling different scenarios for how high sea levels will rise by 2100, based on how well humanity succeeds in mitigating the climate crisis.

I'll make a prediction, too: I predict that, in 2100, if things aren't as bad as the worst-case scenario, some people will gleefully point at the prediction from 2023 and say, "See? Look! Things aren't nearly as bad as they predicted. We did all that for nothing!" And then they'll book themselves a scuba trip to the ruins of Old New Orleans.

“[Where] it’s going to make a huge difference [is] not for us, but for our children’s children. That’s the difficult thing to get your head around.”

And that's why we won't do nearly enough: the people who can actually make a difference only look at quarterly earnings over the next year or so, or the next election in 2-6 years.

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